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Search: id:A005712
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| A005712 |
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Coefficient of x^4 in expansion of (1+x+x^2)^n.
(Formerly M4129)
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+0
10
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| 1, 6, 19, 45, 90, 161, 266, 414, 615, 880, 1221, 1651, 2184, 2835, 3620, 4556, 5661, 6954, 8455, 10185, 12166, 14421, 16974, 19850, 23075, 26676, 30681, 35119, 40020, 45415, 51336, 57816, 64889, 72590, 80955, 90021, 99826, 110409, 121810, 134070
(list; graph; listen)
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OFFSET
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2,2
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COMMENT
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a(n) = A111808(n,4) for n>3. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2005
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-3) is the number of 5-subests of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Oct 03 2007
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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LINKS
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Milan Janjic, Two Enumerative Functions
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Trinomial Coefficient
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FORMULA
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G.f.: (x^2)*(1+x-x^2)/(1-x)^5.
Binomial(n+4,n)+binomial(n+3,n)-binomial(n+2,n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2006
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MAPLE
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[seq(binomial(n+4, n)+binomial(n+3, n)-binomial(n+2, n), n=0..50)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2006
A005712:=(-1-z+z**2)/(z-1)**5; [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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Cf. A000574, A005581, A005714-A005716.
a(n)= A027907(n, 4), n >= 2 (fifth column of trinomial coefficients).
Sequence in context: A061293 A005900 A138357 this_sequence A070893 A027963 A034199
Adjacent sequences: A005709 A005710 A005711 this_sequence A005713 A005714 A005715
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 02 2000
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